Słowa kluczowe:
active disturbance rejection
computed torque control
differential algebraic equations
parallel rehabilitation robot
real-time implementation
system identification

In this report, ankle rehabilitation routines currently approved by physicians are implemented via novel control algorithms on a recently appeared robotic device known as the motoBOTTE. The physician specifications for gait cycles are translated into robotic trajectories whose tracking is performed twofold depending on the availability of a model: (1) if obtained via the Euler-Lagrange approach along with identification of unknown plant parameters, a new computed-torque control law is proposed; it takes into account the parallel-robot characteristics; (2) if not available, a variation of the active disturbance rejection control technique whose parameters need to be tuned, is employed. A detailed discussion on the advantages and disadvantages of the model-based and model-free results, from the continuous-time simulation to the discrete-time implementation, is included.

Przejdź do artykułu
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Słowa kluczowe:
fractional systems
positive systems
the Caputo derivative
controllability
delay
the Metzler matrix

In the paper positive fractional continuous-time linear systems are considered. Positive fractional systems without delays and positive fractional systems with a single delay in control are studied. New criteria for approximate and exact controllability of systems without delays as well as a relative controllability criterion of systems with delay are established and proved. Numerical examples are presented for different controllability criteria. A practical application is proposed.

Przejdź do artykułu
[1] A. Abdelhakim and J. Tenreiro Machado: A critical analysis of the conformable derivative,
*Nonlinear Dynamics*, 95 (2019), 3063–3073, DOI:
10.1007/s11071-018-04741-5.

[2] K. Balachandran, Y. Zhou and J. Kokila: Relative controllability of fractional dynamical systems with delays in control,*Communications in Nonlinear Science and Numerical Simulation*, 17 (2012), 3508–3520, DOI:
10.1016/j.cnsns.2011.12.018.

[3] K. Balachandran, J. Kokila, and J.J. Trujillo: Relative controllability of fractional dynamical systems with multiple delays in control,*Computers and Mathematics with Apllications*, 64 (2012), 3037–3045, DOI:
10.1016/j.camwa.2012.01.071.

[4] P. Duch: Optimization of numerical algorithms using differential equations of integer and incomplete orders, Doctoral dissertation, Lodz University of Technology, 2014 (in Polish).

[5] C. Guiver, D. Hodgson and S. Townley: Positive state controllability of positive linear systems.*Systems and Control Letters*, 65 (2014), 23–29, DOI:
10.1016/j.sysconle.2013.12.002.

[6] R.E. Gutierrez, J.M. Rosario and J.T. Machado: Fractional order calculus: Basic concepts and engineering applications, Mathematical Problems in Engineering, 2010 Article ID 375858, DOI: 10.1155/2010/375858.

[7] T. Kaczorek:*Positive 1D and 2D Systems, Communications and Control Engineering*, Springer, London 2002.

[8] T. Kaczorek: Fractional positive continuous-time linear systems and their reachability,*International Journal of Applied Mathematics and Computer Science*, 18 (2008), 223–228, DOI:
10.2478/v10006-008-0020-0.

[9] T. Kaczorek: Positive linear systems with different fractional orders,*Bulletin of the Polish Academy of Sciences: Technical Sciences*, 58 (2010), 453–458, DOI:
10.2478/v10175-010-0043-1.

[10] T. Kaczorek: Selected Problems of Fractional Systems Theory,*Lecture Notes in Control and Information Science*, 411, 2011.

[11] T. Kaczorek: Constructability and observability of standard and positive electrical circuits,*Electrical Review*, 89 (2013), 132–136.

[12] T. Kaczorek: An extension of Klamka’s method of minimum energy control to fractional positive discrete-time linear systems with bounded inputs,*Bulletin of the Polish Academy of Sciences: Technical Sciences*, 62 (2014), 227–231, DOI:
10.2478/bpasts-2014-0022.

[13] T. Kaczorek: Minimum energy control of fractional positive continuoustime linear systems with bounded inputs,*International Journal of Applied Mathematics and Computer Science*, 24 (2014), 335–340, DOI:
10.2478/amcs-2014-0025.

[14] T. Kaczorek and K. Rogowski:*Fractional Linear Systems and Electrical Circuits*, Springer, Studies in Systems, Decision and Control, 13 2015.

[15] T. Kaczorek: A class of positive and stable time-varying electrical circuits,* Electrical Review*, 91 (2015), 121–124. DOI:
10.15199/48.2015.05.29.

[16] T. Kaczorek: Computation of transition matrices of positive linear electrical circuits,*BUSES – Technology, Operation, Transport Systems*, 24 (2019), 179–184, DOI:
10.24136/atest.2019.147.

[17] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo:*Theory and Applications of Fractional Differential Equations*, North-Holland Mathematics Studies, 204, 2006.

[18] J. Klamka:*Controllability of Dynamical Systems*, Kluwer Academic Publishers, 1991.

[19] T.J.Machado,V. Kiryakova and F. Mainardi: Recent history of fractional calculus,*Communications in Nonlinear Science and Numerical Simulation*, 6 (2011), 1140–1153, DOI:
10.1016/j.cnsns.2010.05.027.

[20] K.S. Miller and B. Ross:*An Introduction to the Fractional Calculus and Fractional Differential Calculus*, Villey, 1993.

[21] A. Monje, Y. Chen, B.M. Viagre, D. Xue and V. Feliu:*Fractional-order Systems and Controls. Fundamentals and Applications*, Springer-Verlag, 2010.

[22] K. Nishimoto:*Fractional Calculus: Integrations and Differentiations of Arbitrary Order*, University of New Haven Press, 1989.

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[28] B. Sikora: Controllability criteria for time-delay fractional systems with a retarded state,* International Journal of Applied Mathematics and Computer Science*, 26 (2016), 521–531, DOI:
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[29] B. Sikora and J. Klamka: Constrained controllability of fractional linear systems with delays in control,*Systems and Control Letters*, 106 (2017), 9–15, DOI:
10.1016/j.sysconle.2017.04.013.

[30] B. Sikora and J. Klamka: Cone-type constrained relative controllability of semilinear fractional systems with delays,*Kybernetika*, 53 (2017), 370–381, DOI:
10.14736/kyb-2017-2-0370.

[31] B. Sikora: On application of Rothe’s fixed point theorem to study the controllability of fractional semilinear systems with delays,*Kybernetika*, 55 (2019), 675–689, DOI:
10.14736/kyb-2019-4-0675.

[32] T. Schanbacher: Aspects of positivity in control theory,*SIAM J. Control and Optimization*, 27 (1989), 457–475.

[33] B. Trzasko: Reachability and controllability of positive fractional discretetime systems with delay,*Journal of Automation Mobile Robotics and Intelligent Systems*, 2 (2008), 43–47.

[34] J. Wei: The controllability of fractional control systems with control delay,*Computers and Mathematics with Applications*, 64 (2012), 3153–3159, DOI:
10.1016/j.camwa.2012.02.065.

Przejdź do artykułu
[2] K. Balachandran, Y. Zhou and J. Kokila: Relative controllability of fractional dynamical systems with delays in control,

[3] K. Balachandran, J. Kokila, and J.J. Trujillo: Relative controllability of fractional dynamical systems with multiple delays in control,

[4] P. Duch: Optimization of numerical algorithms using differential equations of integer and incomplete orders, Doctoral dissertation, Lodz University of Technology, 2014 (in Polish).

[5] C. Guiver, D. Hodgson and S. Townley: Positive state controllability of positive linear systems.

[6] R.E. Gutierrez, J.M. Rosario and J.T. Machado: Fractional order calculus: Basic concepts and engineering applications, Mathematical Problems in Engineering, 2010 Article ID 375858, DOI: 10.1155/2010/375858.

[7] T. Kaczorek:

[8] T. Kaczorek: Fractional positive continuous-time linear systems and their reachability,

[9] T. Kaczorek: Positive linear systems with different fractional orders,

[10] T. Kaczorek: Selected Problems of Fractional Systems Theory,

[11] T. Kaczorek: Constructability and observability of standard and positive electrical circuits,

[12] T. Kaczorek: An extension of Klamka’s method of minimum energy control to fractional positive discrete-time linear systems with bounded inputs,

[13] T. Kaczorek: Minimum energy control of fractional positive continuoustime linear systems with bounded inputs,

[14] T. Kaczorek and K. Rogowski:

[15] T. Kaczorek: A class of positive and stable time-varying electrical circuits,

[16] T. Kaczorek: Computation of transition matrices of positive linear electrical circuits,

[17] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo:

[18] J. Klamka:

[19] T.J.Machado,V. Kiryakova and F. Mainardi: Recent history of fractional calculus,

[20] K.S. Miller and B. Ross:

[21] A. Monje, Y. Chen, B.M. Viagre, D. Xue and V. Feliu:

[22] K. Nishimoto:

[23] K.B. Oldham and J. Spanier:

[24] I. Podlubny: Fractional Differential Equations:

[25] S.G. Samko, A.A. Kilbas and O.I. Marichev:

[26] J. Sabatier, O.P. Agrawal and J.A. Tenreiro Machado: Advances in Fractional Calculus, In:

[27] B. Sikora: Controllability of time-delay fractional systems with and without constraints,

[28] B. Sikora: Controllability criteria for time-delay fractional systems with a retarded state,

[29] B. Sikora and J. Klamka: Constrained controllability of fractional linear systems with delays in control,

[30] B. Sikora and J. Klamka: Cone-type constrained relative controllability of semilinear fractional systems with delays,

[31] B. Sikora: On application of Rothe’s fixed point theorem to study the controllability of fractional semilinear systems with delays,

[32] T. Schanbacher: Aspects of positivity in control theory,

[33] B. Trzasko: Reachability and controllability of positive fractional discretetime systems with delay,

[34] J. Wei: The controllability of fractional control systems with control delay,

The Bearingless Switched Reluctance Motor (BSRM) is a new technology motor, which overcomes the problems of maintenances required associated with mechanical contacts and lubrication of rotor shaft effectively. In addition, it also improves the output power developed and rated speed. Hence, the BSRM can achieve high output power and super high speed with less size and cost. It has a considerable ripple in the net-torque due to its critical non-linearity and the salient pole structures of both stator and rotor poles. The resultant torque ripple, especially in these motors, causes the more vibrations and acoustic noises will affects the levitated rotor safety also. Practically at high-speed operations, the accurate measurement of the rotor position is complicated for conventional mechanical sensors. A new square currents control with global sliding mode control based sensorless torque observer is proposed to minimize the torque ripple and achieve a smooth, robust operation without using any mechanical sensors. The proposed controller is designed based on the error between the reference and measured torque values. The sliding mode torque observer measures the torque from the actual phase voltages, currents, and look-up tables. The simulation model has been modelled to validate the proposed methodology. From the simulation outputs, it is clear that the reduction of torque ripple by the proposed method shows improved than the conventional sliding mode controller. The overall system is more robust to the external disturbances, and it also gets efficient torque profile.

Przejdź do artykułu
Słowa kluczowe:
discrete-time systems
periodic systems
affine systems
uniform global asymptotic stabilization
state feedback

Affine discrete-time control periodic systems are considered. The problem of global asymptotic stabilization of the zero equilibrium of the closed-loop system by state feedback is studied. It is assumed that the free dynamic system has the Lyapunov stable zero equilibrium. The method for constructing a damping control is extended from time-invariant systems to time varying periodic affine discrete-time systems. By using this approach, sufficient conditions for uniform global asymptotic stabilization for those systems are obtained. Examples of using the obtained results are presented.

Przejdź do artykułu
Słowa kluczowe:
hyperchaos
hyperchaotic systems
hidden attractors
multistability
sliding mode control
circuit design

In this work, we have developed a new 4-D dynamical system with hyperchaos and hidden attractor. First, by introducing a feedback input control into the 3-D Ma chaos system (2004), we obtain a new 4-D hyperchaos system with no equilibrium point. Thus, we derive a new hyperchaos system with hidden attractor. We carry out an extensive bifurcation analysis of the newhyperchaos model with respect to the three parameters.We also carry out probability density distribution analysis of the new hyperchaotic system. Interestingly, the new nonlinear hyperchaos system exhibits multistability with coexisting attractors.Next,we discuss global hyperchaos selfsynchronization for the newhyperchaos system via Integral Sliding Mode Control (ISMC). As an engineering application, we realize the new 4-D hyperchaos system with an electronic circuit via MultiSim. The outputs of the MultiSim hyperchaos circuit show good match with the numerical MATLAB plots of the hyperchaos model. We also analyze the power spectral density (PSD) of the hyperchaos of the state variables using MultiSim.

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Słowa kluczowe:
estimation
Kalman filter
adaptive PD filter
filter amplification

The article presents the algorithm that enables adaptive determination of the amplification coefficient in the filter equation provided by Kalman. The method makes use of an estimation error, which was defined for this purpose, and its derivative to determine the direction of correction changes of the gain vector. This eliminates the necessity to solve Riccati equation, which causes reduction of the method computational complexity. The experimental studies carried out using the proposed approach relate to the estimation of state coordinates describing river pollution using the BOD (biochemical oxygen demand) and DO (dissolved oxygen) indicators).The acquired results indicate that the suggested method does better estimations than the Kalman filter. Two indicators were used to measure the quality of estimates: the Root Mean Squared Error (RMSE) and the Mean Percentage Error (MPE).

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Słowa kluczowe:
fractional-order systems
D-stability
recursive algorithms
complex polynomials
root locus
symmetries
control-theory didactics

It is shown how a stability test, alternative to the classical Routh test, can profitably be applied to check the presence of polynomial roots inside half-planes or even sectors of the complex plane. This result is obtained by exploiting the peculiar symmetries of the root locus in which the basic recursion of the test can be embedded. As is expected, the suggested approach proves useful for testing the stability of fractional-order systems. A pair of examples show how the method operates. It is believed that the suggested geometric approach can also be of some didactic value in introducing basic control-system tools to engineering students.

Przejdź do artykułu
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[32] A. Lienard and M.H. Chipart: Sur le signe de la partie réelle des racines d’une équation algébrique,*Journal of Mathématiques Pures et Appliquée,* 10(6), (1914), 291–346.

[33] M. Marden:*Geometry of Polynomials *[2nd ed.], American Mathematical Society, Providence, RI, USA, 1966.

[34] I. Petras: Stability of fractional-order systems with rational orders: a survey,*Fractional Calculus & Applied Analysis*, 12(3), (2009), 269–298.

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[36] E.J. Routh:*A Treatise on the Stability of a Given State of Motion, Particularly Steady Motion*, Macmillan, London, UK, 1877.

[37] J. Schur: Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind,*Journal für die reine und angewandte Mathematik, *147, (1917) 205– 232, DOI:
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[38] R.Tempo: A Simple Test for Schur Stability of a Diamond of Complex Polynomials,*Proceedings of the 28th IEEE Confewrence on Decision and Control *(1989), 1892–1895.

[39] U. Viaro: Stability tests revisited, In*A Tribute to Antonio Lepschy*, G. Picci and M.E. Valcher, Eds., Edizioni Libreria Progetto, Padova, Italy, pp. 189– 199, 2007.

[40] U. Viaro:*Twenty–Five Years of Research with Antonio Lepschy*, Edizioni Libreria Progetto, Padova, Italy, 2009.

[41] U. Viaro (preface by W. Krajewski):*Essays on Stability Analysis and Model Reduction*, Polish Academy of Sciences, Warsaw, Poland, 2010.

[42] R.S. Vieira: Polynomials with symmetric zeros,*arXiv:*1904.01940v1 [math.CV], 2019.

Przejdź do artykułu
[2] A.T. Azar, A.G. Radwan, and S.Vaidyanathan, Eds.:

[3] R. Becker, M. Sagraloff. V. Sharma, J. Xu, and C. Yap: Complexity analysis of root clustering for a complex polynomial,

[4] T.A. Bickart and E.I. Jury: The Schwarz–Christoffel transformation and polynomial root clustering,

[5] Y. Bistritz: Optimal fraction–free Routh tests for complex and real integer polynomials,

[6] D. Casagrande, W. Krajewski, and U. Viaro: On polynomial zero exclusion from an RHP sector,

[7] D. Casagrande, W. Krajewski, and U. Viaro: Fractional-order system forced-response decomposition and its application, In

[8] A. Cohn: Über die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise,

[9] Ph. Delsarte and Y. Genin: The split Levinson algorithm,

[10] Ph. Delsarte and Y. Genin: On the splitting of classical algorithms in linear prediction theory,

[11] A. Doria–Cerezo and M. Bodson: Root locus rules for polynomials with complex coefficients,

[12] A. Doria–Cerezo and M. Bodson: Design of controllers for electrical power systems using a complex root locus method,

[13] A. Ferrante, A. Lepschy, and U. Viaro: A simple proof of the Routh test,

[14] A. Hurwitz: Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt,

[15] R. Imbach and V.Y. Pan: Polynomial root clustering and explicit deflation,

[16] E.I. Jury and J. Blanchard: A stability test for linear discrete systems in table form,

[17] T. Kaczorek:

[18] W. Krajewski, A. Lepschy, G.A. Mian, and U. Viaro: A unifying frame for stability-test algorithms for continuous-time systems,

[19] W. Krajewski, A. Lepschy, G.A. Mian, and U. Viaro: Common setting for some classical z-domain algorithms in linear system theory,

[20] W. Krajewski and U. Viaro: Root locus invariance: Exploiting alternative arrival and departure points,

[21] B.C. Kuo:

[22] P.K.Kythe:

[23] A. Lepschy, G.A. Mian, and U. Viaro: A stability test for continuous systems,

[24] A. Lepschy, G.A. Mian, and U. Viaro: A geometrical interpretation of the Routh test,

[25] A. Lepschy, G.A. Mian, and U. Viaro: Euclid-type algorithm and its applications,

[26] A. Lepschy, G.A. Mian, andU. Viaro: Splitting of some s-domain stabilitytest algorithms,

[27] A. Lepschy, G.A. Mian, and U. Viaro: An alternative proof of the Jury- Marden stability criterion,

[28] A. Lepschy, G.A. Mian, and U. Viaro: Efficient split algorithms for continuous-time and discrete-time systems,

[29] A. Lepschy and U. Viaro: On the mechanism of recursive stability-test algorithms,

[30] A. Lepschy and U. Viaro: Derivation of recursive stability-test procedures,

[31] S. Liang, S.G. Wang, and Y. Wang: Routh-type table test for zero distribution of polynomials with commensurate fractional and integer degrees,

[32] A. Lienard and M.H. Chipart: Sur le signe de la partie réelle des racines d’une équation algébrique,

[33] M. Marden:

[34] I. Petras: Stability of fractional-order systems with rational orders: a survey,

[35] A.G. Radwan, A.M. Soliman, A.S. Elwakil, and A. Sedeek: On the stability of linear systems with fractional order elements,

[36] E.J. Routh:

[37] J. Schur: Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind,

[38] R.Tempo: A Simple Test for Schur Stability of a Diamond of Complex Polynomials,

[39] U. Viaro: Stability tests revisited, In

[40] U. Viaro:

[41] U. Viaro (preface by W. Krajewski):

[42] R.S. Vieira: Polynomials with symmetric zeros,

Słowa kluczowe:
infinite dimensional systems
approximate controllability
Green’s function approach
flexible Kirchhoff–Love plate

In the paper approximate controllability of second order infinite dimensional system with damping is considered. Applying linear operators in Hilbert spaces general mathematical model of second order dynamical systems with damping is presented. Next, using functional analysis methods and concepts, specially spectral methods and theory of unbounded linear operators, necessary and sufficient conditions for approximate controllability are formulated and proved. General result may be used in approximate controllability verification of second order dynamical system using known conditions for approximate controllability of first order system. As illustrative example using Green function approach approximate controllability of distributed dynamical system is also discussed.

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Słowa kluczowe:
structural control system
vibration
simple fractions
synthesis
rational functions
dynamic characteristics

The paper formulates and formalises a method for selecting parameters of the tuned mass damper (TMD) for primary systems with many degrees of freedom. The method presented uses the properties of positive rational functions, in particular their decomposition, into simple fractions and continued fractions, which is used in the mixed method of synthesis of vibrating mechanical systems. In order to formulate a method of tuning a TMD, the paper discusses the basic properties of positive rational functions. The main assumptions of the mixed synthesis method is presented, based on which the general method of determining TMD parameters in the case of systems with many degrees of freedom was formulated. It has been shown that a tuned mass damper suppresses the desired resonance zone regardless of where the excitation force is applied. The advantages of the formulated method include the fact of reducing several forms of the object’s free vibration by attaching an additional system with the number of degrees of freedom corresponding to the number of resonant frequencies reduced. In addition, the tuned mass damper determined in the case of excitation force applied at a single point can be attached to any element of the inertial primary system without affecting the reduction conditions in this way. It results directly from the methodology formalised in the paper. As part of the paper, numerical calculations were performed regarding the tuning of the TMD to the first form of free vibration of a system with 3 degrees of freedom. The parameters determined were subjected to analysis and verification of the correctness of the calculations carried out. For the considered case of a system with 3 degrees of freedom together with a TMD, time responses of displacement, from each floor, were generated to excitation induced by a harmonic force equal to the first form of vibration of the basic system. In addition, in the case of the parameters obtained, the response of the inertial element system to which the TMD was attached to random white noise excitation was determined.

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Słowa kluczowe:
multi-criteria decision-making
interval-valued dual hesitant fuzzy elements
Archimedean t-conorm and t-norm
prioritized weighted averaging operator
prioritized weighted geometric operator

Multi-criteria decision making (MCDM) technique and approach have been a trending topic in decision making and systems engineering to choosing the probable optimal options. The primary purpose of this article is to develop prioritized operators to multi-criteria decision making (MCDM) based on Archimedean t-conorm and t-norms (At-CN&t-Ns) under interval-valued dual hesitant fuzzy (IVDHF) environment. A new score function is defined for finding the rank of alternatives in MCDM problems with IVDHF information based on priority levels of criteria imposed by the decision maker. This paper introduces two aggregation operators: At-CN&t-N-based IVDHF prioritized weighted averaging (AIVDHFPWA), and weighted geometric (AIVDHFPWG) aggregation operators. Some of their desirable properties are also investigated in details. A methodology for prioritization-based MCDM is derived under IVDHF information. An illustrative example concerning MCDM problem about a Chinese university for appointing outstanding oversea teachers to strengthen academic education is considered. The method is also applicable for solving other real-life MCDM problems having IVDHF information.

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